Formula for success: Multilevel modelling of Formula One Driver and Constructor performance, 1950-2014

This is a repository copy of

Formula for success: Multilevel modelling of Formula One

Driver and Constructor performance, 1950-2014

.

White Rose Research Online URL for this paper:

http://eprints.whiterose.ac.uk/96995/

Version: Supplemental Material

Article:

Bell, A. orcid.org/0000-0002-8268-5853, Smith, J., Sabel, C.E. et al. (1 more author)

(2016) Formula for success: Multilevel modelling of Formula One Driver and Constructor

performance, 1950-2014. Journal of Quantitative Analysis in Sports, 12 (2). pp. 99-112.

ISSN 1559-0410

https://doi.org/10.1515/jqas-2015-0050

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Online appendix to

Formula for Success: Multilevel modelling of Formula

One driver and constructor performance, 1950-2014

.

This appendix provides further figures and tables that could not be included in the printed version of the paper, but which may nonetheless be of interest to some readers. The contents of this document are as followed:

 Table A1: shows the sensitivity of the models to different outcome variable transformations, as

mentioned in section 3.3.2.

 Table A2: shows the insignificance of the race-level and driver-year level random effects (as mentioned in

section 3.1.

 Table A3: shows basic null models in the form of equation 2 in the paper, including the model that

produced Figure 1 and the variances mentioned in section 4.2.

 Table A4: shows complex models in the form of equation 3 in the paper, including the model that

produced Figure 2.

 Table A5: shows complex models that produced differential random effects for different weather

conditions, including the model that produces Figure 3.

 Table A6: shows complex models in the form of equation 7 in the paper, including the model that

produced Figure 4.

 Table A7: shows the separately coded version of the models shown in table A5.

 Table A8: shows the separately coded version of the models shown in table A6.

 Table A9: shows a list of the top 50 drivers, extending Figure 1.

 Table A10: shows predicted and actual champions in each season, as mentioned in section 4.3.

 Table A11: shows predicted and actual champions for the 2015 season, as mentioned in section 4.3.

 Figure A1: a visual representation of the team, team-year and driver variances from the model in table

A3.

 Figure A2: shows all the drivers in a single graph

 Figure A3: shows the top 20 team-level residuals

 Figure A4: shows the top 20 team-year-level residuals

 Figure A5: shows the top 20 drivers under different weather conditions, as mentioned in section 4.1.

Table A1: Level 1 residual plots, and predicted top 10 drivers, according to a variety of different dependent variables and model specifications.

Y variable L1 Normal Q-Q plot L1 Residuals Histogram Top 10 drivers

Rankit of Points scored 1. Fangio 2. Clark 3. Prost 4. Alonso 5. Stewart 6. Senna 7. Schumacher 8. Vettel 9. E Fittipaldi 10. Piquet

Table A2: Model results for 5 and 6 level model including a driver-year level and a race level (for a model with 500,000 iterations).

6-level model 5-level model

Estimate 95% CIs ESS Estimate 95% CIs ESS

Fixed Part

Constant 0.434 0.403 0.464 19157 0.434 0.404 0.464 19492 Ndrivers -gm -0.042 -0.046 -0.038 150153 -0.042 -0.046 -0.038 153110 NewComp -gm 0.977 0.343 1.614 166553 0.978 0.346 1.611 168164

Random Part

Driver-Year

Variance 0.003 0.001 0.006 984 0.003 0.001 0.006 1007 Race Variance 0.000 0.000 0.000 7314

Team Variance 0.023 0.016 0.032 32087 0.023 0.016 0.031 32399 Team-Year Variance 0.021 0.017 0.025 29449 0.021 0.017 0.025 28677 Driver Variance 0.013 0.010 0.017 32126 0.013 0.010 0.017 31736 Level 1 Variance 0.288 0.282 0.294 349694 0.288 0.282 0.294 350524

DIC: 33786.549 33762.679

Table A3: Models showing variance partitioning, controlling for competitiveness and the number of drivers.

(a) 1979-2014 (b) 1950-2014 - Schumacher as 1 driver (produces Figure 1)

(c) 1950-2014 - Schumacher as 2 drivers

Estimate 95% CIs Estimate 95% CIs Estimate 95% CIs

Fixed Part

Constant 0.065 -0.011 0.139 0.434 0.403 0.464 0.433 0.403 0.463 Ndrivers -gm -0.049 -0.060 -0.039 0.986 0.353 1.615 0.982 0.350 1.615 Comp -gm 1.343 0.047 2.641 -0.042 -0.046 -0.038 -0.042 -0.046 -0.038

Random Part

Team Variance 0.066 0.040 0.102 0.022 0.015 0.031 0.023 0.016 0.032 Team-Year Variance 0.042 0.034 0.051 0.022 0.018 0.026 0.021 0.018 0.025 Driver Variance 0.017 0.012 0.025 0.013 0.010 0.017 0.013 0.010 0.017 Level 1 Variance 0.468 0.457 0.480 0.289 0.284 0.295 0.289 0.283 0.295

Table A4: Models with variance as a function of Year, 1979-2014

(a) Year Fixed Effect Only (b) Year effect random at Team level (c) Year effect random at Team-Year level (d) Year effect random at Driver level

(e) Year effect random at all higher levels

(produces figure 2)

Estimate 95% CIs Estimate 95% CIs Estimate 95% Cis Estimate 95% CIs Estimate 95% CIs

Fixed Part

Cons 0.065 -0.012 0.140 0.063 -0.017 0.138 0.066 -0.011 0.141 0.064 -0.012 0.137 0.065 -0.013 0.141

Ndrivers -gm -0.052 -0.063 -0.040 -0.052 -0.063 -0.041 -0.052 -0.063 -0.040 -0.052 -0.064 -0.041 -0.053 -0.064 -0.042

Comp -gm 1.564 0.203 2.927 1.657 0.311 3.009 1.569 0.204 2.934 1.482 0.124 2.839 1.634 0.292 2.976

Year -gm -0.002 -0.006 0.002 -0.002 -0.008 0.004 -0.001 -0.005 0.003 -0.002 -0.006 0.002 -0.002 -0.008 0.005

Random Part

Team Level

Cons 0.067 0.041 0.104 0.058 0.033 0.094 0.068 0.041 0.105 0.062 0.037 0.098 0.059 0.034 0.095

Covariance 0.001 -0.000 0.002 0.001 -0.000 0.002

Year 0.000 0.000 0.000 0.000 0.000 0.000

Team-Year Level

Cons 0.042 0.034 0.051 0.037 0.030 0.045 0.041 0.033 0.050 0.032 0.021 0.043 0.028 0.018 0.038

Covariance 0.000 - 0.000 0.001 0.000 -0.000 0.001

Year 0.000 0.000 0.000 0.000 0.000 0.000

Driver Level

Cons 0.017 0.012 0.025 0.016 0.011 0.023 0.013 0.007 0.020 0.017 0.012 0.024 0.012 0.007 0.019

Covariance 0.000 -0.001 0.000 - 0.000 -0.001 - 0.000

Year 0.000 0.000 0.000 0.000 0.000 0.000

Level 1 Var 0.468 0.457 0.480 0.468 0.457 0.480 0.468 0.457 0.479 0.468 0.457 0.480 0.468 0.457 0.479

Table A5: Models with variance as a function of weather (dry/wet conditions), 1979-2014

(a) Weather Fixed effects only

(b) Weather random at Team Level

(c) Weather random at Team-Year Level

(d) Weather random at Driver Level

(e) Weather random at all higher levels (produces figure 3)

Estimate 95% CIs Estimate 95% CIs Estimate 95% CIs Estimate 95% CIs Estimate 95% CIs

Fixed Part

Cons 0.067 -0.010 0.140 0.060 -0.019 0.136 0.066 -0.010 0.141 0.065 -0.011 0.139 0.060 -0.019 0.136

Ndrivers -gm -0.049 -0.060 -0.039 -0.049 -0.060 -0.039 -0.050 -0.060 -0.039 -0.049 -0.060 -0.039 -0.050 -0.060 -0.039

Comp -gm 1.345 0.051 2.639 1.350 0.050 2.654 1.353 0.064 2.648 1.347 0.053 2.647 1.357 0.062 2.654

Wet 0.002 -0.032 0.035 0.040 -0.010 0.092 0.003 -0.033 0.038 0.013 -0.025 0.051 0.042 -0.010 0.096

Random Part

Team Level

Cons 0.066 0.040 0.102 0.071 0.044 0.109 0.065 0.040 0.101 0.065 0.040 0.102 0.070 0.044 0.108

Covariance -0.015 -0.030 -0.003 -0.013 -0.028 0.000

Wet 0.010 0.004 0.022 0.009 0.003 0.020

Team-Year Level

Cons 0.042 0.034 0.051 0.042 0.034 0.051 0.044 0.035 0.054 0.042 0.034 0.051 0.044 0.035 0.053

Covariance -0.008 -0.018 0.000 -0.007 -0.016 0.001

Wet 0.011 0.003 0.026 0.010 0.003 0.023

Driver Level

Cons 0.017 0.012 0.025 0.018 0.012 0.025 0.017 0.012 0.025 0.019 0.013 0.026 0.019 0.013 0.026

Covariance -0.005 -0.010 0.000 -0.003 -0.009 0.002

Wet 0.005 0.001 0.012 0.004 0.001 0.010

Level 1 Var 0.468 0.457 0.480 0.467 0.456 0.479 0.467 0.456 0.478 0.468 0.457 0.479 0.466 0.455 0.477

Table A6: Models with variance as a function of track type (permanent/temporary/street), 1979-2014

(a) Track type Fixed Effects only

(b) Track type random at Team level

(c) Track type random at Team-Year

level (d) Track type random at Driver level

(e) Street and temp random at all higher levels (produces figure 4)

Estimate 95% CIs

Estimat

e 95% CIs Estimate 95% CIs Estimate 95% CIs Estimate 95% CIs

Fixed Part

Cons 0.066 -0.010 0.140 0.059 -0.021 0.136 0.067 -0.009 0.141 0.064 -0.013 0.139 0.062 -0.018 0.139

Ndrivers -gm -0.049 -0.060 -0.039 -0.049 -0.059 -0.039 -0.050 -0.060 -0.039 -0.049 -0.060 -0.039 -0.049 -0.060 -0.039

Comp -gm 1.342 0.041 2.638 1.354 0.059 2.652 1.342 0.048 2.639 1.317 0.010 2.613 1.339 0.041 2.634

Temp 0.002 -0.043 0.047 0.026 -0.034 0.088 0.002 -0.046 0.051 0.014 -0.038 0.066 0.027 -0.038 0.092

Street 0.000 -0.033 0.033 0.022 -0.018 0.064 0.001 -0.033 0.036 0.012 -0.031 0.055 0.016 -0.029 0.062

Random Part

Team Level

Cons 0.066 0.040 0.102 0.074 0.047 0.114 0.064 0.039 0.099 0.066 0.041 0.103 0.072 0.045 0.111

temp/cons -0.011 -0.028 0.004 -0.009 -0.025 0.005

Temp 0.010 0.003 0.023 0.008 0.002 0.019

street/cons -0.009 -0.020 -0.002 -0.005 -0.014 0.001

street/temp 0.002 -0.002 0.007 0.001 -0.001 0.005

Street 0.003 0.001 0.008 0.001 0.000 0.004

Team-Year Level

Cons 0.042 0.034 0.051 0.042 0.034 0.051 0.047 0.038 0.057 0.042 0.034 0.051 0.045 0.037 0.055

temp/cons -0.005 -0.018 0.007 -0.002 -0.015 0.010

Temp 0.036 0.013 0.068 0.029 0.010 0.058

street/cons -0.015 -0.024 -0.007 -0.011 -0.020 -0.004

street/temp 0.010 -0.001 0.024 0.005 -0.003 0.016

Street 0.011 0.004 0.023 0.006 0.002 0.015

Driver Level

Cons 0.017 0.012 0.025 0.017 0.012 0.024 0.018 0.012 0.025 0.019 0.013 0.026 0.018 0.012 0.025

temp/cons -0.005 -0.012 0.002 -0.002 -0.010 0.005

Temp 0.014 0.005 0.030 0.011 0.004 0.025

street/cons -0.004 -0.011 0.002 -0.002 -0.009 0.004

street/temp 0.011 0.002 0.022 0.009 0.001 0.019

Street 0.019 0.009 0.034 0.017 0.007 0.030

Level 1 _{Var 0.468 0.457 0.480 0.468 0.457 0.479 0.465 0.454 0.476 0.465 0.454 0.477 0.463 0.452 0.474}

Table A7: Separately coded model with variance as a function of weather, 1950-2014

Weather with separate coding

Estimate 95% CIs

Fixed Part

Cons 0.458 0.42 0.494

Ndrivers -gm 0.997 0.364 1.63

Comp -gm -0.042 -0.046 -0.038

Wet -0.029 -0.06 0.002

Random Part

Team Level

Dry 0.025 0.017 0.034

Covariance 0.02 0.013 0.029

Wet 0.018 0.011 0.028

Team-Year Level

Dry 0.022 0.019 0.027

Covariance 0.019 0.014 0.024

Wet 0.02 0.013 0.029

Driver Level

Dry 0.014 0.011 0.018

Covariance 0.012 0.008 0.016

Wet 0.012 0.007 0.018

Level 1 _{Variance 0.288 0.282 0.294}

Table A8: Separately coded model with variance as a function of track type, 1950-2014

Track type with separate coding

Estimate 95% CIs

Fixed Part

Cons 0.431 0.399 0.462

Ndrivers -gm 0.969 0.339 1.602

Comp -gm -0.042 -0.046 -0.038

Temporary 0.018 -0.035 0.071

Street 0.011 -0.02 0.043

Random Part

Team level

Permanent 0.026 0.018 0.036

Perm/Temp Cov 0.022 0.014 0.033

Temporary 0.025 0.014 0.041

Perm/Street Cov 0.021 0.014 0.03

Street/Temp Cov 0.019 0.012 0.029

Street 0.02 0.013 0.029

Team-Year level

Permanent 0.024 0.019 0.028

Perm/Temp Cov 0.021 0.014 0.028

Temporary 0.037 0.021 0.056

Perm/Street Cov 0.017 0.013 0.021

Street/Temp Cov 0.019 0.012 0.028

Street 0.016 0.011 0.023

Driver level

Permanent 0.014 0.01 0.018

Perm/Temp Cov 0.011 0.006 0.017

Temporary 0.016 0.008 0.029

Perm/Street Cov 0.013 0.01 0.018

Street/Temp Cov 0.013 0.007 0.02

Street 0.018 0.012 0.025

Level 1 _{Variance 0.287 0.281 0.293}

Table A9: Top 50 drivers based on the driver level residuals from model A1d (Michael Schumacher treated as two drivers, pre 2006 and post 2010)

Rank Driver Residual Rank Driver Residual

1 Juan Manuel Fangio 0.333 26 Robert Kubica 0.129

2 Alain Prost 0.300 27 Carlos Reutemann 0.128

3 Michael Schumacher (pre-2006) 0.286 28 Tom Pryce 0.128

4 Jim Clark 0.276 29 Stirling Moss 0.123

5 Ayrton Senna 0.265 30 Martin Brundle 0.121

6 Fernando Alonso 0.263 31 Rubens Barrichello 0.119

7 Nelson Piquet 0.238 32 Daniel Ricciardo 0.119

8 Jackie Stewart 0.232 33 Alan Jones 0.119

9 Emerson Fittipaldi 0.217 34 Kimi Raikkonen 0.118

10 Sebastian Vettel 0.213 35 Patrick Depailler 0.118

11 Christian Fittipaldi 0.198 36 Carlos Pace 0.117

12 Lewis Hamilton 0.175 37 Richie Ginther 0.116

13 Graham Hill 0.169 38 Denny Hulme 0.115

14 Dan Gurney 0.166 39 Thierry Boutsen 0.113

15 Jody Scheckter 0.165 40 Mike Hawthorn 0.111

16 Jenson Button 0.160 41 Jean-Pierre Beltoise 0.106

17 Marc Surer 0.158 42 Heinz-Harald Frentzen 0.105

18 Damon Hill 0.157 43 Prince Bira 0.102

19 Louis Rosier 0.143 44 Keke Rosberg 0.100

20 Elio de Angelis 0.141 45 Clay Regazzoni 0.098

21 Ronnie Peterson 0.140 46 Luigi Fagioli 0.097

22 Nino Farina 0.130 47 Jack Brabham 0.093

23 Nick Heidfeld 0.130 48 Jacques Villeneuve 0.093

24 Pedro Rodríguez 0.129 49 Nico Rosberg 0.092

Table A10: Comparison between predictions of the champion (from the model as in equation 3) and the actual champion, for years 1979-2014. Schumacher is treated as two drivers.

Year Model's predicted champion Actual Position of model's champion Actual Champion Model's position of actual champion Notes

1979 J Scheckter 1st J Scheckter 1st 1980 C Reutemann 3rd A Jones 2nd 1981 C Reutemann 2nd N Piquet 3rd 1982 Keke Rosberg 1st K Rosberg 1st

1983 A Prost 2nd N Piquet 7th Piquet beaten in the model by 2nd-4th place drivers (Prost, Arnoux and Tambay, who were within 20 points of him), Jonathan Palmer (who only raced one race and outperformed his Williams team-mate in that race), Jacques Laffite (who benefits in the model compared to the championship because he missed two races, and Keke Rosberg, who won the championship the previous year.

1984 A Prost 1st A Prost 1st 1985 A Prost 1st A Prost 1st

1986 N Piquet 3rd A Prost 2nd Only 3 points between 1st and 3rd in the championship

1987 N Piquet 1st N Piquet 1st

1988 A Prost 2nd A Senna 2nd Only 3 points between 1st and 2nd in the championship; Very close in model predictions between 1st and second as well

1989 A Prost 1st A Prost 1st 1990 A Senna 1st A Senna 1st 1991 A Senna 1st A Senna 1st

1992 R Patrese 2nd N Mansell 3rd Very small differences in the model predictions of 1st and 3rd. Mansell gained lots of 1st places, so won by a long way in points (but there is a less clear gap in finishing position, reducing the advantage when points are transformed).

1993 A Prost 1st A Prost 1st

1994 A Senna Not classified M Schumacher 3rd Senna only raced 3 races, finishing none (the third race was, tragically, his last). However because he didn't race in many races, it doesn't count against him or his team-year too much. Thus, his high driver residual was weighted heavily in his favour.

1995 M Schumacher 1st M Schumacher 1st 1996 D Hill 1st D Hill 1st

1997 M Schumacher 2nd/DSQ J Villeneuve 4th Schumacher was 2nd (3 points behind Villeneuve) but was disqualified from the final standings for dangerous driving.

1998 M Schumacher 2nd M Hakkinen 2nd

1999 M Schumacher 5th M Hakkinen 3rd Schumacher only completed seven races when he broke his leg, at which point he was second in the championship

2005 F Alonso 1st F Alonso 1st

2006 M Schumacher 2nd F Alonso 2nd Model produces a close result between Alonso and Schumacher. In the championship there was only a 13 point difference.

2007 F Alonso 3rd K Raikkonen 2nd Only 1 point between 1st and 3rd on the championship

2008 L Hamilton 1st L Hamilton 1st 2009 J Button 1st J Button 1st

2010 F Alonso 2nd S Vettel 2nd Only 4 points between 1st _{and 2}nd _{in the championship}

2011 S Vettel 1st S Vettel 1st

2012 F Alonso 2nd S Vettel 2nd Only 3 points between 1st _{and 2}nd _{in the championship}

2013 S Vettel 1st S Vettel 1st 2014 L Hamilton 1st L Hamilton 1st

Table A11: Out of sample predictions for the 2015 F1 season. Team year residuals are assumed not to change from 2014, and all trends are extrapolated. Based on a model including random slopes on year, using data from 1979. Michael Schumacher is treated as two drivers.

2015 Actual Results Driver 2015 team Predicted Ranking

1 Lewis Hamilton Mercedes 5

2 Nico Rosberg Mercedes 6

3 Sebastian Vettel Ferrari 2

4 Kimi Räikkönen Ferrari 4

5 Valtteri Bottas Williams 12

6 Felipe Massa Williams 13

7 Daniil Kvyat Red Bull 9

8 Daniel Ricciardo Red Bull 7

9 Sergio Pérez Force India 16

10 Nico Hülkenberg Force India 14

11 Romain Grosjean Lotus 11

14 Pastor

Maldonado Lotus 10 16 Jenson Button McClaren 3

17 Fernando Alonso McClaren 1

18 Marcus Ericsson Sauber 15

21 Will Stevens Marussia 17

F

A P

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-

N

M

S

-

B

A

Figure A3: Top 20 team-level residuals, based on model (b) in table A3

F

A P

-

-

N

M

S

-

1

_F

_{A P}

-

-

N

M

S

B

A

1

These graphs (figures A6 and A7) were produced by mo the model, to allow effects and their uncertainty to be most easily computed. These models produce exactly